Maybe of interest to your project: I wind my clapton coils with this machine: the Avidartisan Daedalus pro v2 microwinder:
It has a very small chuck to hold small diameter wire (in your case, a core) and a spool holder for the thinner wire to be winded onto it. It doesn’t have a counter - since it’s meant to create lengths of vaping coil wire rather than induction coils - but you might be able to repurpose it.
At any rate, for the price, you can’t go wrong.